Connecting Euclidean to light-cone correlations: from flavor nonsinglet in forward kinematics to flavor singlet in non-forward kinematics

We present a unified framework for the perturbative factorization connecting Euclidean correlations to light-cone correlations. Starting from nonlocal quark and gluon bilinear correlators, we derive the relevant hard-matching kernel up to the next-to-leading-order, both for the flavor singlet and non-singlet combinations, in non-forward and forward kinematics, and in coordinate and momentum space. The results for the generalized distribution functions (GPDs), parton distribution functions (PDFs), and distribution amplitudes (DAs) are obtained by choosing appropriate kinematics. The renormalization and matching are done in a state-of-the-art scheme. We also clarify some issues raised on the perturbative matching of GPDs in the literature. Our results provide a complete manual for extracting all leading-twist GPDs, PDFs as well as DAs from lattice simulations of Euclidean correlations in a state-of-the-art strategy, either in coordinate or in momentum space factorization approach.


Introduction
The partonic structure of hadrons plays a crucial role in mapping out their 3D image and in describing the experimental data collected at high energy colliders such as the Large Hadron Collider (LHC) and the Electron-Ion Collider (EIC).The simplest quantities that characterize such structure include the leading-twist collinear parton distribution functions (PDFs), generalized parton distributions (GPDs) and distribution amplitudes (DAs).They are all defined by collinear parton operators and are nonperturbative in nature.A lot of efforts have been devoted to determining them from fitting various experimental data for both singlet and non-singlet sectors (see, e.g., [1][2][3][4] and references therein).Among them, the experimental extraction of the leading-twist GPDs is much more difficult due to their complicated kinematic dependence.Some progress has been made on reducing the uncertainties in perturbative QCD [5][6][7][8] and from power corrections [9][10][11][12][13].
In the past few years, we have also witnessed rapid developments on extracting these quantities from lattice QCD (see [14][15][16][17] for a recent review) based on various proposals [18][19][20][21][22][23][24][25][26].Among them, one of the widely used options is to start from the quasi-light-front (quasi-LF) correlators [15,20,21], which are equal-time quark and gluon correlators defined on a Euclidean space interval.It can be connected to the light-front (LF) correlators defining the collinear parton distributions, either through a short-distance factorization [24] in coordinate space or through a large-momentum factorization [15,20,27] in momentum space.Such a factorization requires a hard coefficient function or matching kernel as the input, which is, in general, different for different hadronic functions.
There have been many studies on the perturbative matching relevant to the computation of collinear parton distributions (for a summary, see Refs.[15,17]), most of which are focused on the flavor-nonsinglet combination due to its simplicity.The state-of-the-art is the next-to-next-to-leading order (NNLO) calculation for the isovector unpolarized quark PDFs [28,29].However, the majority of the results cannot be directly implemented in lattice calculations.This is because, on the one hand, dimensional regularization (DR) and MS scheme are often used in such calculations, while these schemes cannot be realized in lattice calculations.Therefore, a conversion between the lattice renormalization scheme to the MS scheme needs to be performed before the matching can be applied.On the other hand, certain schemes are regularization-independent [30][31][32][33], and thus avoid the scheme conversion mentioned above.But in such schemes the renormalization factors, when applied in the large-momentum factorization approach, introduce undesired infrared (IR) contributions at large distances, and thus invalidate the factorization formula.In Ref. [34], a hybrid renormalization scheme has been proposed to circumvent this issue, and used in subsequent lattice calculations of PDFs and DAs [35][36][37][38][39][40][41].
In contrast to the flavor-nonsinglet combination, the flavor-singlet quark and gluon combinations have been much less studied [42][43][44][45][46][47] in the literature.In Refs.[44,45], the results for the unpolarized flavor-singlet quark and gluon PDFs have been presented, where a discrepancy was found in the unphysical region for the matching kernel of the unpolarized gluon PDF.In Ref. [48], the one-loop matching for quark GPDs has been recalculated using a subtraction approach, in which a discrepancy from previous calculations [49,50] was found in the flavor-nonsinglet combination.In the same paper, certain mixing contributions between the flavor-singlet quark GPDs and the gluon GPDs have also been considered.However, a complete calculation of both coordinate and momentum space factorizations in the state-of-the-art renormalization scheme is still missing.
Given that the PDFs, DAs and GPDs are all defined by the same collinear parton operators, it is highly desirable to develop a unified computational framework for all of them, so that the perturbative matching for each can be obtained by a choice of appropriate kinematics.It is the purpose of the present paper to develop such a framework.We start from nonlocal quark and gluon bilinear operators, and first present a general coordinate space factorization formula.The relevant coordinate space matching kernels are then calculated for general non-forward kinematics, both for the flavor singlet and non-singlet combina-tions, where the renormalization and matching are done in a state-of-the-art scheme.We also clarify some of the observed discrepancies between results by different groups.Our results provide a complete manual for a state-of-the-art extraction of all leading-twist GPDs, PDFs as well as DAs from lattice simulations, either in coordinate or in momentum space factorization approach.Our framework will also greatly facilitate higher-order perturbative calculations involving collinear parton correlators of spacelike separation, which are important for extracting collinear parton structure functions from lattice calculations to a high accuracy.
The rest of the paper is organized as follows: In Sec. 2, we give the operator definitions used throughout the paper, and present a factorization between quasi-LF and LF correlators that is valid in general kinematics, both in coordinate space and in momentum space.In Sec. 3, we show the calculation of the matching kernel for flavor-singlet quark and gluon combinations in non-forward kinematics, and give the results in the MS, ratio and hybrid schemes.We then discuss in Sec. 4 the special kinematic limits in which the PDFs and DAs are recovered, and present the corresponding matching kernels in all schemes.Sec. 5 is our conclusion.Some technical details of the calculation are summarized in the Appendix.

Factorization between Euclidean and light-front correlations
In this section, we give the definition of the relevant quasi-LF correlators, and present the factorization formula connecting them to the LF correlators defining collinear parton distributions.

Operator bases
To begin with, let us consider the following spatial non-local quark correlator where Γ is the Dirac structure taken as γ t , γ z γ 5 , γ t γ ⊥ γ 5 in the present paper.They are used to define the unpolarized, longitudinally polarized (helicity) and transversely polarized (transversity) quark quasi-LF correlations, respectively.In the discussion below, we will denote them by subscripts u, h, t, respectively.z 1 , z 2 are 4-vectors along the spatial z direction, and [z 1 , z 2 ] represents the straight Wilson line in the fundamental representation ensuring gauge invariance of the operator, and P indicating the path-ordering, where ϵ µνρσ is the Levi-Civita tensor with ϵ 0123 = −ϵ 0123 = 1.Flavor indices of the quark fields are omitted for brevity.
The flavor non-singlet and singlet quark combinations are related to the following combinations of quark correlators (summation over quark flavors will be omitted hereafter for simplicity) where the upper sign corresponds to the unpolarized (vector) case while the lower sign corresponds to the helicity (axial-vector) case.Note that the operator defining the quark transversity is chiral-odd and thus does not mix with gluons, and the combination of the corresponding quark correlators always takes "+" sign.Analogously, we can introduce the following spatial nonlocal operators for gluons in d spacetime dimensions, where ) with F µν being the gluon field strength tensor.The Wilson line [z 1 , z 2 ] in this case resides in the adjoint representation.Ŝ indicates a symmetrization over the indices and a trace subtraction in the operator that follows.µ, ν, α represent the d − 2 dimensional transverse indices in DR.In lattice simulations, d should be replaced by 4. In O g,u the summation over µ, ν can also be taken over all Lorentz components (i.e., including the longitudinal and time directions), which generates identical leading-twist LF distributions.We also consider this operator choice and list the results in the Appendix.Note that Bose symmetry indicates that the gluon operators must satisfy certain relations under z 1 ↔ z 2 .For example, O g,h (z 1 , z 2 ) shall be anti-symmetric, whereas O g,{u,t} (z 1 , z 2 ) shall be symmetric under the exchange z 1 ↔ z 2 .

Quasi-light-front correlations
By sandwiching the nonlocal parton operators (2.4) and (2.5) in different external hadronic states, we can define the quasi-LF correlations (qLFCs) relevant for the calculation of GPDs, PDFs and DAs.For example, the quasi-GPDs are defined through the following non-forward matrix element where the hadron state is denoted by its momentum and spin as The forward quasi-PDF limits are obtained when the initial and final states are identical.Quasi-DAs are hadron-to-vacuum matrix elements of the same operators, and thus can also be obtained from the above matrix element by taking appropriate kinematic limits.The general factorization formula of quasi-LF correlators can be summarized into a matrix form as where we have suppressed the factorization scale dependence for notational simplicity.l.t.stands for the leading-twist projection of the non-local correlator which acts as the generating function of leading-twist local operators [51][52][53][54][55][56].The existence of the local operator product expansion naturally implies the factorization formula above.⊗ stands for the convolution, and h.t.denotes higher-twist terms.The 2 × 2 mixing matrix C is the perturbative matching function in coordinate space, and for the non-singlet combination only C qq is non-vanishing.In general, the mixing matrix depends on two Feynman parameters α, β and the invariant distance squared z 2 12 , and the convolution takes the form of a double integral.For example, writing down the first row of Eq. (2.7) explicitly, we have dβ C qq (α, β, µ 2 z 2 vanish at one-loop, they will contribute starting from two-loop level.The factorization in Eq. (2.7) is valid to all orders in perturbation theory.In this work, we lay the foundation of the computational framework and focus on the mixing matrix up to the next-to-leadingorder (NLO), and leave higher-order calculations to future publications.
The leading-twist operators O l.t. on the r.h.s. of Eq. (2.7) define the leading-twist parton distributions when sandwiched between suitable external hadronic states.For example, from the LF quark operator O l.t. γ − being a lightlike vector, we define the leading-twist unpolarized quark GPDs H and E [53,57,58] with the kinematic variables (2.10) The skewness parameter ξ differs in the qLFCs and LF correlations only by power-suppressed contributions [59].We will not distinguish them in the following.The definitions of all leading-twist parton distributions (GPDs, PDFs, DAs) can be found in Refs.[60,61].
We would like to stress that the matching kernels for the GPDs reduce to that of the PDFs and DAs in certain kinematic limits.Thus, in the following, we will begin with the generic non-forward matrix elements relevant to the GPDs.The PDF and DA limits will then be discussed as special cases.

Pseudo-distributions
One way to extract collinear parton distributions, or more precisely their moments, from the qLFCs is to use the short-distance factorization [33].Keeping z 2  12 fixed and Fourier transforming the quasi-LF distance (or Ioffe-time) ζ i = z i P z (i = 1, 2) to the momentum fraction, one obtains the so-called pseudo-distributions.We begin with the pseudo-GPDs (pGPDs) * , which are related to the non-forward matrix element defined in the previous subsection via the following Fourier transform where N is a normalization factor.For the unpolarized, longitudinally and transversely polarized quark pGPDs N = 1/(2P t ), 1/(2P z ) and 1/(2P t S ⊥ ) respectively, while for gluon pGPDs N = 1 regardless of the polarization.The pGPDs can be factorized into the GPDs as follows with the 2 × 2 matching matrix where we follow the notation of Ref. [61], τ 1 , τ 2 , y 1 and y 2 are momentum fractions that can be expressed in a more symmetric way as The matching kernels in coordinate-and pseudo-space are related by the Fourier transform (2.11), which leads to the following formula that allows us to get the latter from the former directly, where a delta function enforcing the momentum conservation τ 1 + τ 2 = y 1 + y 2 is omitted.Two formulas prove to be helpful for computing the integral above.we include them in the Appendix together with some technical details.It is worth pointing out that the symmetry relations for the diagonal and off-diagonal elements are recovered automatically from the integral in (2.14) following the operator definitions (2.4), (2.5) as expected.

Quasi-distributions
In the large-momentum factorization or large-momentum effective theory (LaMET) [15,20,21] approach, one defines the quasi-distribution as a Fourier transform of the qLFCs with respect to the spatial interval.For the purpose of deriving the matching kernel for quasi-distributions, we can either Fourier transform the qLFCs directly to momentum space, or first to pseudo-space and then do a double Fourier transform to momentum space as adopted in Refs.[62,63].Both strategies yield the same result, but the second one turns out to be easier, so we use the second strategy for our calculation.The quasi-GPDs (qGPDs) are then related to the pGPDs as and the factorization formula for the qGPDs reads with the matching coefficient in momentum space (2.17) Since the z 12 -dependence in the pGPDs only appears through logarithmic functions, the z i -integrals in (2.15) can be computed in a procedural manner [62].
3 Matching for flavor-singlet quarks and gluons in non-forward kinematics In the following, we calculate the matching kernel for non-forward qLFCs or for the GPDs.The calculation can be greatly simplified by observing that the four-momentum transfer t = (P 1 − P 2 ) 2 only leads to power suppressed contributions, and thus do not contribute to the leading-twist matching.Therefore, we assume t = 0 in the calculation hereafter.
Another simplification comes from that the matching kernel is the same for the GPDs defined by the same nonlocal operator [59].Therefore, we do not need to consider all the GPDs for the calculation.Throughout the calculation, we use DR (d = 4−2ϵ) to regularize both ultraviolet (UV) and infrared (IR) divergences.This also simplifies the calculation as scaleless integrals will vanish.We will first present the matching kernels in the MS scheme, and then in the ratio and hybrid schemes.

Matching kernel for quasi-light-front correlations
The matching matrix for the non-forward qLFCs can be calculated by replacing the hadron states with quark or gluon parton states and expanding the factorization formula up to the NLO.The resulting expressions are summarized in Table 1, where |q⟩, |q ′ ⟩ and |g⟩, |g ′ ⟩ denote different on-shell quark and gluon external states with momenta p µ i = (p 0 i = p z i , 0, 0, p z i ) with i = 1, 2. The superscripts (0) and (1) denote the tree-level and NLO quantities, respectively.The factorization formula ensures that the IR divergence cancels out in the gluon C (1) Table 1.The NLO matching matrix for non-forward qLFCs.
numerator of these expressions so that they are IR finite.More generally, the formulas in the MS scheme can be written schematically as where i, j ∈ q, g and the superscript R (B) stands for renormalized (bare) quantities.The nonlocal (bare) collinear operators O B i receive quantum corrections that are computable in perturbative QCD.The multiplicative UV renormalization constants Z UV remove the UV singularities and are known to three loops already [64].The IR singularities, on the other hand, are removed by the renormalization constants of the specific LF operators.
Notice that we have inserted 1 = Z −1 O j ⊗ Z O j to simultaneously remove the IR singularities present in both the bare coefficient function C B ij and the LF operator O B j .The fact that the known renormalization constants Z UV and Z O j in, e.g., [61,65,66] completely remove all 1/ϵ poles provide strong checks for our matching coefficients.

C qq
The calculation of the matrix entry C qq is relatively simple, and has already been calculated in momentum space at one-loop in Refs.[48][49][50]59].Here we give the results for the qLFCs, with the one-loop Feynman diagrams being shown in Fig. 1.In the Feynman gauge, the ladder diagram yields the following contribution where a s = α s /(4π) and η = −(z 2 12 µ 2 e γ E )/4 in the MS scheme.The calculation is done with the collinear on-shell condition of the quark momenta p 2 1 = p 2 2 = 0, p 1 // p 2 † .Note that only the ladder diagram depends on the Dirac structure in the quark operator.The contribution of the vertex diagram and its conjugate are independent of the Dirac structure, and takes the following form The Wilson line self-energy diagram yields Since we do not distinguish UV and IR divergences, the quark self-energy diagram vanishes in DR.Also the one-loop matrix element of O l.t.q gives zero.Adding up all the contributions, we obtain the one-loop matching kernel for the quark qLFCs in the MS scheme where L z = ln 4e −2γ E −µ 2 z 2

12
. Throughout this work, we set µ R = µ F = µ for convenience with µ R and µ F being the renormalization and factorization scale, respectively.The "+" subscript denotes the usual plus-prescription for the singularity

C qg
Next we consider C qg , which requires calculating the gluon matrix element of the quark operator, and is non-zero only for the unpolarized and helicity operators.The corresponding diagram is shown in Fig. 2.This diagram gives ) † More generally, p1 and p2 are understood to be the parton momenta, and are related to the hadron momentum as p z i = yiP z i .Here we use pi as a short-hand notation in intermediate steps.where T = {g ρσ ⊥ , ϵ ρσ ⊥ } correspond to the unpolarized and helicity case, respectively.We have applied z 12 • A = p 1 • A = p 2 • A = 0 as our external gluons are physical on-shell states.Since gauge invariance guarantees that the matching function is convoluted with the whole O g operator, it is sufficient to extract the coefficient function that convolutes with z α 12 ∂ α A µ which is part of F z 12 µ .In the lightcone limit, we have F z 12 µ ∝ F +µ recovering the LF gluon GPD.The cross diagram I 2,qg can be obtained from I 1,qg by interchanging z 1 ↔ z 2 and multiplying with an overall factor −1. Taking into account that gluons are bosons and making change of variables, we find Thus, we get the final expression of the qLFCs in the MS scheme Putting everything together, we are able to write down the matching kernel as (3.9)

C gq
The calculation of C gq requires calculating the quark matrix element of the gluon operator, and the corresponding diagram is shown in Fig. 3.The Feynman diagram yields where Γ = {γ z , γ z γ 5 } denotes the generated structure in the unpolarized and helicity case, respectively.The cross diagram I 2,gq is obtained by z 1 ↔ z 2 and multiplying with an overall factor −1. Adding them up, the total contribution reads From the above result, we obtain the matching kernel in the MS scheme as (3.12)

C gg
The calculation of C gg requires calculating the gluon matrix element of the gluon operator.
The corresponding Feynman diagrams are depicted in Fig. 4. The ladder diagram gives where T is the same as before for the unpolarized and helicity operators, and Ŝ has been defined in Eq. (2.5).The second diagram is These two diagrams are the only diagrams that depend on the choice of operators.The other diagrams yield the same result for unpolarized, helicity and transversity operators apart from the generated tree-level structure (indicated by { T , Ŝ} below) (3.15)I 3,gg contains the conjugate diagram which can be obtained by making the replacement z 1 ↔ z 2 , p 1 ↔ p 2 and α → 1 − α in the contribution of the third diagram.In addition, F σz 12 (z 2 ) can be expanded as following The next three diagrams give To proceed, we need to recast I 4,gg , I 6,gg into a form dictated by the factorization theorem, or in other words, to a form similar to I 1,gg .This can be done by using some integration tricks.For example, the term A ρ (p 1 )A σ (p 2 ) can be written as the form of two Feynman variables α and β,

.18)
I 7 contains scaleless integrals only, as can be seen after writing down the explicit amplitude Thus, it vanishes in DR.The Wilson-line self-energy can be directly written down, Adding all contributions together, including the conjugate diagrams, and using integrationby-parts (IBP) techniques and boson symmetry (symmetry under p 1 ↔ p 2 and z 1 ↔ z 2 ), we find that all structures with the phase factor e −i(p 1 +p 2 )•z α 12 cancel out, as required by the factorization theorem.Now we are able to extract the total one-loop coefficient function, which reads The above calculations present the complete one-loop matching kernels in coordinate space that appear in the factorization of quark and gluon quasi-LF correlations in nonforward kinematics.In Ref. [67], the authors provide the one-loop correction for C qq,u which serves as a useful check for our result.

Pseudo-GPDs and quasi-GPDs
In this subsection, we present the matching kernels for the pGPDs and qGPDs.

Matching kernel in pseudo space
Following the discussion in previous subsections, the one-loop matching kernels of the pGPDs can be obtained from that in coordinate space by a Fourier transform in Eq. (2.11), and take the following form in the MS scheme: qq,h = C (1)  qq,u + 2 The singularity as y i → x i gets regulated by the plus-prescription where the integral limits are determined by the range of the momentum fractions in the LF distributions.
Note that the Fourier transform on the C gq channel (3.12) can be most easily done by writing e ix 1 z 12 dx 1 , assuming z 12 > 0 , (3.25) where the lower integral limit ∞(−1 + iϵ) is chosen for convenience.Such a choice is, however, arbitrary and therefore brings ambiguities.The ambiguities are removed by requiring that the matching coefficients in the pseudo and coordinate space must generate identical Mellin moments (for more details, see discussions in Appendix A).
In the following, we list the results for C qg , C gq and C gg : • Quark in gluon with qg,h = • Gluon in quark with • Gluon in gluon with

Matching kernel in momentum space
The matching kernels for qGPDs in momentum space takes the following form in the MS scheme: • Quark in quark with qq,h = C (1)  qq,u + 2 • Quark in gluon with qg,h = • Gluon in quark with • Gluon in gluon In these formulas, we have defined We have checked that all evolution kernels in the above results are in agreement with Ref. [61].In addition, our results for C qq,{h,t} , C qg,{h,t} are consistent with those in Ref. [48], while the results for C (1) qg,u are slightly different from those of Ref. [48] due to different choices of quark operators.It is worth pointing out that one kinematic region is missing in the early calculation of the matching kernels in Refs.[49,50,59] due to an incomplete analytical continuation.To see this, we can temporarily assume y > 0 in Eq. (3.33), and do an explicit expansion of C (1) qq according to the regions of momentum fractions, then we have (by switching to the notation of Refs.[49,50,59] and assuming ξ > 0) C (1)  qq (x, y, ξ) where the region θ(−ξ < x < ξ)θ(x > y) is missing in the kinematic setup in Refs.[49,50,59], and shall be recovered through a correct analytical continuation from the results there.

Ratio and hybrid schemes
So far our discussion has been focused on DR and MS scheme.However, on the lattice where the lattice spacing acts as a UV regulator, a different renormalization scheme has to be adopted.A convenient choice in the literature is the ratio renormalization [33], which is based on the fact that the quark and gluon quasi-LF operators are multiplicatively renormalized [30,64,[68][69][70][71][72][73], thus all UV divergences in the bare qLFCs can be removed by dividing by the correlation of the same operator in a zero-momentum hadronic state.
The ratio scheme does not depend on the regularization, therefore, the matching in this scheme can be conveniently obtained from the results in DR and MS scheme by dividing by the corresponding zero momentum matrix element result.The ratio scheme matching can be directly applied to lattice renormalized qLFCs (after taking the continuum limit of the latter) at short distances.However, the ratio renormalization cannot be directly used in the LaMET factorization approach.In this approach, one needs to do a Fourier transform to momentum space,

PDF and DA limits
In this section, we present the matching kernels for the PDFs and DAs that emerge from the results in previous sections by taking special kinematic limits.

PDFs
In coordinate space, the factorization formula for the forward qLFCs defining the PDFs takes the form h(z 12 , p z , µ) with the following matching kernels • Quark in quark • Quark in gluon • Gluon in quark • Gluon in gluon
In the literature, the short-distance factorization is often carried out in coordinate space instead of the pseudo space.Therefore, we skip the pseudo-PDFs here and consider the momentum space factorization for the quasi-PDFs (qPDFs) only.The complete factorization for quark and gluon PDFs takes the form ‡ q x, µ P z = Since we have assumed t = 0 throughout our calculation, the forward limit can be simply obtained by taking the skewness ξ → 0. The matching kernels of qPDFs are obtained from that of qGPDs through It is clear that no reduction formula for gluon transversity case in a spin-one-half hardon.
For hardons with spin one or higher, gluon transversity is visible in the forward limit [60,61] The results for the singlet PDFs are summarized below.
• Quark in quark qq,t , F • Quark in gluon • Gluon in quark • Gluon in gluon T , F T , F T , F where Several remarks are in order.First of all, our results for F qq are consistent with those in Ref. [75].For the mixing matching coefficients F (1) qg , our result for the polarized case is consistent with that in Ref. [48], while for the unpolarized case there is a slight difference because we have chosen a different Dirac structure in the quark bilinear operator.Secondly, our ratio scheme result for the unpolarized gluon PDF is in agreement with that derived in Eq. (7.28) of Ref. [45], which also starts from coordinate space and then Fourier transforms to momentum space.However, both the results of Ref. [45] and our results show a discrepancy with earlier calculations performed directly in momentum space [44] in the unphysical region.In the ratio scheme, the contribution of the unphysical region in momentum space comes entirely from the Fourier transform of the ln z 2

DAs
Finally, we consider the kinematic limit where the DAs are recovered.The quark operators with Γ = γ z γ 5 , γ t , γ t γ ⊥ γ 5 that we introduce at the beginning define the leading Fock state quasi-DA in a pseudoscalar, longitudinally polarized vector, and transversely polarized vector meson, respectively.We denote them by subscripts p, h and t.In coordinate space, the factorization formula for the DAs takes the same form as that for the GPDs, and the matching kernels are the same as shown in Eq. (3.4).In momentum space, the factorization formula becomes φ x, µ P z = 1 0 dy V x, y; µ P z ϕ(y, µ), (4.13)where the matching kernel V (x, y) can be obtained from that for the qGPDs by taking the limit V x, y; µ P z = C x, 1 − x, y, 1 − y; µ P z , ( where the quark and antiquark momentum fractions are given by x (y) and 1 − x (1 − y), respectively.For completeness, we summarize the results of the matching kernel below, V  qq,t , V (1)  qq,p = V where we have defined

Conclusion
In this paper, we have developed a unified framework for perturbative calculations of the hard-matching kernel connecting collinear qLFCs to the LF correlations.We started by deriving the matching kernel for flavor-singlet quark and gluon correlations in non-forward kinematics in DR and MS scheme in coordinate space.The results for the GPDs, PDFs and DAs were obtained by choosing appropriate kinematics.We then converted our results to the state-of-the-art ratio and hybrid schemes, and gave the matching kernel both in coordinate and in momentum space.Our results provide a complete manual for a state-ofthe-art extraction of all leading-twist GPDs, PDFs as well as DAs from lattice simulations, both for the flavor-singlet and nonsinglet combinations, and either in coordinate or in momentum space factorization approach.Our framework has the potential to greatly facilitate higher-order perturbative calculations involving collinear qLFCs.We will push this forward and present the NNLO results in forthcoming publications.
Note added: While this paper is being finalized, another preprint [78] appears which calculates the one-loop matching for gluon quasi-GPDs in momentum space with a completely different approach.We find that the results agree up to terms that vanish upon convolution with the lightcone distributions (and thus have no impact on physical results), except for the mixing term C gq which we believe is due to the prescription used in dealing with the additional pole 1/z 12 that compensates for the mismatch in the mass dimensions of the gluon and quark fields.While our prescription ensures correct Mellin moments to all orders, the prescription of [78] does not.We will investigate this issue in a future publication.

Figure 1 .
Figure 1.One-loop corrections to C qq .The conjugate diagram to the second one and the quark self-energy diagram are not shown.

Figure 2 .
Figure 2. One-loop correction to C qg .The cross diagram is not shown.

Figure 3 .
Figure 3. One-loop correction to C gq .The cross diagram is not shown.

8 Figure 4 .
Figure 4. One-loop corrections to C gg .The conjugate diagrams and self-energy diagrams are not shown.I 2 represents the contribution of four-gluon vertex.

. 10 )
‡ Note that we follow the conventions in Ref.[61] throughout.Other definitions for the LF-GPDs/PDFs may result in factorization formulas of slightly different forms with xg(x) our work→ g(x)others .